The chromatic convergence theorem and a tower in algebraic $K$-theory

In this note we show how the chromatic convergence theorem of Hopkins and Ravenel implies that a tower of relative algebraic K-theories constructed by Waldhausen converges to the p-local part of the algebraic K-theory of the one-point space relative to the K-theory of the integers. The notion of convergence used here is made precise using the language of pro-homotopy theory.